90 ANALYTICAL MECHANICS Since YI and v2 are in the same line, their difference will be a vector in the same direction. Therefore in this particular case the acceleration is constant not only in magnitude but also in direction. The following definition of acceleration is general and holds true whatever the manner in which the velocity changes. The magnitude of the acceleration of a particle at any point of its path equals the time rate at which its velocity changes at the instant it occupies that point. The analytical expression for this definition may be obtained by a reasoning similar to that employed in deriving the analytical definition of velocity. Sup- F pose it is required to find the acceleration at P (Fig. 56). Let YI and v2 denote the velocities at two neighboring points PI and P2. Then the ratio r V2 ~ Vl gives the average rate at which the velocity changes during the interval of time t, which it takes the particle to move from Pi to P2. Therefore f is the average acceleration for that interval of time. In general this average acceleration will not be the same as the acceleration at P. But by taking PI and P2 nearer and nearer to P the difference between the average acceleration and the required acceleration may be made as small as desired. Therefore at the limit when PI, P, and P2 become successive positions of the particle, the average acceleration becomes identical with the acceleration at P, and the last equation takes the form f--v- (VIII)